Tuning fractional order proportional differential controllers based on vector in the complex plane for second order systems

A new method for automatic adaptive optimization of strain estimation in phase-sensitive optical coherence tomography (OCT) is introduced. More specifically, this paper focuses on optimizing the estimation of strain using the vector method, in which OCT signals are treated as vectors in the complex plane. In phase-sensitive optical coherence elastography, the tissue strain is extracted from the interframe phase variation between the complex-valued scans acquired for the initial and deformed tissue. This phase variation is proportional to interframe displacements of scatterers and corresponds to the argument of the pixel-by-pixel product of the initial OCT scan and complex-conjugate deformed scan. Measurement noises and the so-called ‘speckle noise’ that are intrinsic to OCT scans cause degradation of the derived scan obtained by such multiplication. To mitigate the noise influence, complex-valued pixel amplitudes in the derived scan are usually averaged over a certain window. The interframe strain is found by estimating the gradient of the interframe phase difference. The noise influence on the strain estimation can also be reduced by increasing the scale over which the phase-variation gradient is estimated. However, choosing a too large window for preliminary averaging may significantly distort the reconstructed strain distribution. Similarly, a too large scale for gradient estimation may also cause errors in the estimated-strain magnitude and even its sign (because of possible phase wrapping). Therefore, appropriately performed adaptive choice of the strain-estimation parameters can greatly improve the quality of strain estimation. Here, we present a unified approach for adaptive choice of both the averaging-window size and gradient-estimation scale that were initially considered separately. The new method is computationally simplified but enables approximately the same strain-estimation quality, as demonstrated using both simulated and experimental OCT data.

The PID control strategy has been widely used in today's industrial production and daily life. At the same time, this control strategy is also the most indispensable part of the undergraduate “process control” curriculum. In order to effectively promote the student's intuitive understanding of this strategy, this project enables students to be flexible in the actual control system. Therefore, a virtual simulation based on virtual instrument technology (LabVIEW) and simulation technology (MATLAB) is proposed. A MATLAB program was designed using NI's LABVTEW virtual instrument development platform. By selecting the appropriate options for the design of the LabVIEW front panel, the three parameters Kp, Ki, and Kd of the PID controller of the linear system of the second order can be quickly determined, and the proportional controller of the closed-loop control system under the input step signal can be observed. The response curves of the proportional integral controller and the proportional integral derivative controller are obtained by analyzing the response curves under the step input signal to obtain the control laws of the controllers with different control laws, and the analysis and summary of the three parameters of the second-order system PID are obtained. The general order and the effect on system stability.

The improvement of rescue robots has been vigorously in perilous surroundings for investigate and rescue processes. The secure genus of mobile robots is called snake-like robot. The paper presents implementation of trajectory tracking for snake-like robot by using Proportional controller. In this work, the frame design of snake-like robot model has been accomplished based on the control function with mathematical geometry. The tracking path for snake-like robot was established with the circular path. And then the estimation between the actual tracking and original path could be analysed with MATLAB programming. In this paper, snake-like robot head position is controlled to converge the desired trajectory tracking. In the control system, the rest of the units are automatically converged to track the path of the preceding units by controlling one unit in the head. The tracking error could be reduced by using Proportional controller. The performances of the tracking paths for snake-like robot were shown in this paper.

This paper proposes design of dynamic out-put feedback controller feeding supplementary control signal to FACTS device for improving damping in system. Feedback controller design involves infinity control, model reduction, and closed loop pole-placements in addition to exposure to uncertainties in load conditions on system. A well designed controller satisfying infinity norm on the resulting closed loop system with output feedback meeting multiple objectives from pole-placements inside a conic sector in complex s-plane, robust to changing load conditions on system is attained. A Polytopic representation of system for changing operating point with load conditions called polytopic linear differential inclusion (PLDI) is considered in the design. The Proposed controller design simplifies steps otherwise involved in robust controller design. LMI framework is used in obtaining controller parameters. Controller designed is dynamic and is obtained by reducing the order of the original system to smaller value. The resulting feasible solution yields optimum values of matrix variables and hence a controller parameters both feasible and optimum.

In this chapter, the properties of the transfer function and frequency response of first and second order systems are studied on some examples from electrical circuit laws. We show that their properties are governed by the poles (i.e., the zeros of the denominator) of the transfer function which is a rational fraction. A geometric argument based on the location of the poles of the transfer function in the complex plane allows a qualitative interpretation of the behavior of the frequency response with varying frequency. This geometric interpretation is easily generalized to situations with any number of zeros and poles. It proves useful for the understanding of the general behavior of filters. The study begins here with the simplest system, the first order system. Then the second order circuit system is presented thoroughly. The logarithmic Bode representation of the frequency gain is introduced and its advantages demonstrated. The quality factor Q of a resonant circuit is defined.

It is shown that the flow in the state plane associated with a linear constant-coefficient second-order system may be represented by a circle in the complex plane. The main properties of these circles are examined, and they are shown to possess an additive combination property which should be useful in the synthesis of flows. The representation of higher-order flows may be achieved by orthogonally projecting them on to a suitably chosen set of planes, and representing the projected flows by circles in the complex plane. The additive combination property still holds, so that the results should be useful in the design of multivariable feedback controllers. Forced sinusoidal response is considered in a similar way.

The design of the speed controller is important from the point of view of imparting desired transient and steady-state characteristic to the two quadrant converter controlled separately excited dc motor drive system. The poor design of the controller causes in losses and inefficient operation of the drive. Although many improved PI/PID design methods are proposed, such as root locus, Ziegler-Nichols (Z-N) methods etc., but they have shortcomings such as long testing time and limited control performance. To overcome this difficulty, we make use the theory of D-partition technique. Based on analytically characterizing the D-partition boundaries of the controller parameter space, necessary conditions of the maximum degree of stability are derived. The design of the controller and analysis of a converter controlled closed-loop dc motor drive is presented in this paper work. The effect of different factors like the time constant of proportional speed controller and the motor time constant on drive performance is analyzed. The D-partition technique is used to investigate the interaction of proportional speed time constant, amplifier gain and armature time constant on the stability of two quadrant converter-controlled closed loop separately excited dc motor drive. In this paper work, a procedure has been suggested to find the optimum value of the amplifier gain to give minimum settling time, which is the most important requirement of high performance drives or high fidelity drives. Using the D-partition technique, a comprehensive performance analysis of system design is carried out in this paper, using MATLAB environment.

The paper examines some basic concepts of the quantum theory. It is concluded that through the concepts of rotating vectors in the complex plane and statistical formulation of quantum uncertainty the wave function ψ has its own well defined physical meaning. The approach of the present paper evidences once more the tight link between quantum theory and relativistic theory.

It is commonly recognized that a convenient formulation for problems in planar kinematics is obtained by considering links to be vectors in the complex plane. However, scant attention has been paid to the natural interpretation of complex vectors as isotropic coordinates. These coordinates, often considered a special trick for analyzing four-bar motion, are in fact uniquely suited to two new techniques for analyzing polynomial systems: the BKK bound and the product-decomposition bound. From this synergistic viewpoint, a fundamental formulation of planar kinematics is developed and used to prove several new results, mostly concerning the degree and circularity of the motion of planar linkages. Useful for both analysis and synthesis of mechanisms, the approach both simplifies theoretical proofs and facilitates the numerical solution of mechanism problems.

We have experimentally achieved a gain extinction ratio (GER) of 30 dB at 21.75 dBm pump power in a dual-pump phase sensitive amplifier based on highly nonlinear fiber, which is more efficient than what we predict with the conventional three-wave model. We point out that high-order sidebands play a key role. Then we scrutinize the trajectories of the output signal vector in the complex plane with increasing nonlinear phase shift and understand the roll-over behavior of GER versus pump power curve. We found that the phase "flip" of the output signal occurs around the largest GER. Adjusting design parameters, we show how to improve GER.

An asymptotic approximation for incomplete Gauss sums. II

The effect of nuclear recoil on the elastic scattering of high-energy electrons or muons by zero-spin nuclei is studied by adapting the Breit two-particle Hamiltonian to the case that one of the two particles is of finite size, is spinless, and is nonrelativistic, the other being a normal point Dirac particle. A radial and angular separation of the Dirac equation is still possible. To leading order in the parameter (electron energy)/ (nuclear mass), the effect of the dynamic recoil terms is to rotate the scattering amplitude vectors in the complex plane without changing their magnitudes, a result which is independent of the shape and size of the nuclear charge distribution. To this order, the cross section is affected only by the kinematic recoil corrections. The dynamic recoil terms also influence the scattering amplitudes through terms of order (electron mass)/(nuclear mass). These corrections, owing to large amplification factors in going from phase shifts to cross section, may be of some significance in muon scattering, but are probably of no importance in the analysis of high-energy electron scattering. The dynamic effect is proportional to nuclear charge and therefore nearly as great for heavy as for light nuclei.

In this paper, a tuning procedure of the parameters of fractional order proportional integral controller is presented for time delay system with arbitrary order. On the basis of obtaining stable solution space by utilizing some stability-domain boundaries which include real-root boundary and σ-stability boundary, the optimal solution can be achieved by applying specific frequency-domain specifications which include gain crossing frequency, phase margin, and robustness constraint. Using the same synthesis by traversing phase angle and phase margin in a given interval, the complete solution space of gain crossing frequency can be obtained and visualized, which is equivalent to demonstrate the complete solution space of optimal parameters of fractional order proportional integral controllers. As a comparison, the tuning procedure of the parameters of integral order proportional integral derivative controller is also discussed. At last, the proposed algorithm is validated by numerical simulation and experimental illustrations. The results show the robustness and advantage of the proposed fractional order proportional integral controller over other controllers.

Task Representation and Constraint Analysis for Hybrid Position/Force Controller Design

With the development of factional calculus theory and applications in different fields in recent years, the rational approximation problem of fractional calculus operator has become a hot spot of research. In the early 1950s and 1960s, Carlson and Halijak proposed regular Newton iterating method to implement rational approximation of the one-nth calculus operator. Carlson regular Newton iterating method has a great sense of innovation for the rational approximation of fractional calculus operator, however, it has been used only for certain calculus operators. The aim of this paper is to achieve rational approximation of arbitrary order fractional calculus operator. The realization is achieved via the generalization of Carlson regular Newton iterating method. To construct a rational function sequence which is convergent to irrational fractional calculus operator function, the rational approximation problem of fractional calculus operator is transformed into the algebra iterating solution of arithmetic root of binomial equation. To speed up the convergence, the pre-distortion function is introduced. And the Newton iterating formula is used to solve arithmetic root. Then the approximated rational impedance function of arbitrary order fractional calculus operator is obtained. For nine different operational orders with n changing from 2 to 5, the impedance functions are calculated respectively through choosing eight different initial impedances for a certain operational order. Considering fractional order operation characteristics of the impedance function and the physical realization of network synthesis, the impedance function should satisfy these basic properties simultaneously: computational rationality, positive reality principle and operational validity. In other words, there exists only rational computation of operational variable s in the expression of impedance function. All the zeros and poles of impedance function are located on the negative real axis of s complex plane or the left-half plane of s complex plane in conjugate pairs. The frequency-domain characteristics of impedance function approximate to those of ideal fractional calculus operator over a certain frequency range. Given suitable initial impedance and for an arbitrary operational order, it is proved that the impedance function could meet all properties above by studying the zero-pole distribution and analyzing frequency-domain characteristics of the impedance function. Therefore, the impedance function could take on operational performance of the ideal fractional calculus operator and achieve the physical realization. It is of great effectiveness in the generalization of this kind of method in both theory and experiment. The results educed in this paper are the basis for further theoretic research and engineering application in constructing the arbitrary order fractional circuits and systems.